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- 1. Preface Applications of physics can be found in a wider and
wider range of disciplines in the sci- ences and engineering. It is
therefore more and more important for students, practitioners,
researchers, and teachers to have ready access to the facts and
formulas of physics. Compiled by professional scientists,
engineers, and lecturers who are experts in the day- to-day use of
physics, this Handbook covers topics from classical mechanics to
elementary particles, electric circuits to error analysis. This
handbook provides a veritable toolbox for everyday use in problem
solving, home- work, examinations, and practical applications of
physics, it provides quick and easy access to a wealth of
information including not only the fundamental formulas of physics
but also a wide variety of experimental methods used in practice.
Each chapter contains all the important concepts, formulas, rules
and theorems numerous examples and practical applications
suggestions for problem solving, hints, and cross references M
measurement techniques and important sources of errors as well as
numerous tables of standard values and material properties. Access
to information is direct and swift through the user-friendly
layout, structured table of contents, and extensive index. Concepts
and formulas are treated and presented in a uniform manner
throughout: for each physical quantity dened in the Handbook, its
characteristics, related quantities, measurement techniques,
important formulas, SI-units, transformations, range of
applicability, important relationships and laws, are all given a
unied and compact presentation. This Handbook is based on the third
German edition of the Taschenbuch der Physik published by Verlag
Harri Deutsch. Please send suggestions and comments to the Physics
Editorial Department, Springer Verlag, 175 Fifth Avenue, New York,
NY 10010. Walter Benenson, East Lansing, MI John Harris, New Haven,
CT Horst Stocker, Frankfurt, Germany Holger Lutz, Friedberg,
Germany v
- 2. Contents Preface v Contributors xxiii Part I Mechanics 1 1
Kinematics 3 1.1 Description of motion . . . . . . . . . . . . . .
. . . . . . . . . . . . 3 1.1.1 Reference systems . . . . . . . . .
. . . . . . . . . . . . . . 3 1.1.2 Time . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 8 1.1.3 Length, area, volume . .
. . . . . . . . . . . . . . . . . . . 9 1.1.4 Angle . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 11 1.1.5 Mechanical
systems . . . . . . . . . . . . . . . . . . . . . . 12 1.2 Motion
in one dimension . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 14 1.2.2 Acceleration . . . . . . . . . . . . . . . . . . . . .
. . . . . 17 1.2.3 Simple motion in one dimension . . . . . . . . .
. . . . . . 19 1.3 Motion in several dimensions . . . . . . . . . .
. . . . . . . . . . . . 22 1.3.1 Velocity vector . . . . . . . . .
. . . . . . . . . . . . . . . 23 1.3.2 Acceleration vector . . . .
. . . . . . . . . . . . . . . . . . 25 1.3.3 Free-fall and
projectile motion . . . . . . . . . . . . . . . . . 28 1.4
Rotational motion . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 31 1.4.1 Angular velocity . . . . . . . . . . . . . . . . . .
. . . . . . 32 1.4.2 Angular acceleration . . . . . . . . . . . . .
. . . . . . . . . 33 1.4.3 Orbital velocity . . . . . . . . . . . .
. . . . . . . . . . . . 34 2 Dynamics 37 2.1 Fundamental laws of
dynamics . . . . . . . . . . . . . . . . . . . . . 37 2.1.1 Mass
and momentum . . . . . . . . . . . . . . . . . . . . . 37 2.1.2
Newtons laws . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.1.3 Orbital angular momentum . . . . . . . . . . . . . . . . . .
48 2.1.4 Torque . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 50 2.1.5 The fundamental law of rotational dynamics . . . . .
. . . . 52 vii
- 3. viii Contents 2.2 Forces . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 53 2.2.1 Weight . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 53 2.2.2 Spring torsion
forces . . . . . . . . . . . . . . . . . . . . . 54 2.2.3
Frictional forces . . . . . . . . . . . . . . . . . . . . . . . .
56 2.3 Inertial forces in rotating reference systems . . . . . . .
. . . . . . . . 59 2.3.1 Centripetal and centrifugal forces . . . .
. . . . . . . . . . . 60 2.3.2 Coriolis force . . . . . . . . . . .
. . . . . . . . . . . . . . 62 2.4 Work and energy . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 63 2.4.1 Work . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 63 2.4.2 Energy . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 65 2.4.3 Kinetic
energy . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.4.4
Potential energy . . . . . . . . . . . . . . . . . . . . . . . . 67
2.4.5 Frictional work . . . . . . . . . . . . . . . . . . . . . . .
. 70 2.5 Power . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 70 2.5.1 Efciency . . . . . . . . . . . . . . . . .
. . . . . . . . . . 71 2.6 Collision processes . . . . . . . . . .
. . . . . . . . . . . . . . . . . 72 2.6.1 Elastic straight-line
central collisions . . . . . . . . . . . . . 74 2.6.2 Elastic
off-center central collisions . . . . . . . . . . . . . . 76 2.6.3
Elastic non-central collision with a body at rest . . . . . . . .
76 2.6.4 Inelastic collisions . . . . . . . . . . . . . . . . . . .
. . . . 78 2.7 Rockets . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 79 2.7.1 Thrust . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 79 2.7.2 Rocket equation . . . . . . .
. . . . . . . . . . . . . . . . . 81 2.8 Systems of point masses .
. . . . . . . . . . . . . . . . . . . . . . . 82 2.8.1 Equations of
motion . . . . . . . . . . . . . . . . . . . . . . 82 2.8.2
Momentum conservation law . . . . . . . . . . . . . . . . . 84
2.8.3 Angular momentum conservation law . . . . . . . . . . . . .
85 2.8.4 Energy conservation law . . . . . . . . . . . . . . . . .
. . . 86 2.9 Lagranges and Hamiltons equations . . . . . . . . . .
. . . . . . . . 86 2.9.1 Lagranges equations and Hamiltons
principle . . . . . . . . 86 2.9.2 Hamiltons equations . . . . . .
. . . . . . . . . . . . . . . 89 3 Rigid bodies 93 3.1 Kinematics .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.1.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 93 3.1.2 Center of mass . . . . . . . . . . . . . . . . . . . . .
. . . . 94 3.1.3 Basic kinematic quantities . . . . . . . . . . . .
. . . . . . . 96 3.2 Statics . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 97 3.2.1 Force vectors . . . . . . .
. . . . . . . . . . . . . . . . . . 98 3.2.2 Torque . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 100 3.2.3 Couples . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 101 3.2.4
Equilibrium conditions of statics . . . . . . . . . . . . . . . 103
3.2.5 Technical mechanics . . . . . . . . . . . . . . . . . . . . .
. 104 3.2.6 Machines . . . . . . . . . . . . . . . . . . . . . . .
. . . . 106 3.3 Dynamics . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 111 3.4 Moment of inertia and angular
momentum . . . . . . . . . . . . . . . 111 3.4.1 Moment of inertia
. . . . . . . . . . . . . . . . . . . . . . . 111 3.4.2 Angular
momentum . . . . . . . . . . . . . . . . . . . . . . 116 3.5 Work,
energy and power . . . . . . . . . . . . . . . . . . . . . . . .
118 3.5.1 Kinetic energy . . . . . . . . . . . . . . . . . . . . .
. . . . 119 3.5.2 Torsional potential energy . . . . . . . . . . .
. . . . . . . . 120
- 4. Contents ix 3.6 Theory of the gyroscope . . . . . . . . . .
. . . . . . . . . . . . . . 121 3.6.1 Tensor of inertia . . . . . .
. . . . . . . . . . . . . . . . . . 121 3.6.2 Nutation and
precession . . . . . . . . . . . . . . . . . . . . 124 3.6.3
Applications of gyroscopes . . . . . . . . . . . . . . . . . . 127
4 Gravitation and the theory of relativity 129 4.1 Gravitational
eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.1.1 Law of gravitation . . . . . . . . . . . . . . . . . . . . .
. . 129 4.1.2 Planetary motion . . . . . . . . . . . . . . . . . .
. . . . . 131 4.1.3 Planetary system . . . . . . . . . . . . . . .
. . . . . . . . . 133 4.2 Special theory of relativity . . . . . .
. . . . . . . . . . . . . . . . . 137 4.2.1 Principle of relativity
. . . . . . . . . . . . . . . . . . . . . 137 4.2.2 Lorentz
transformation . . . . . . . . . . . . . . . . . . . . 140 4.2.3
Relativistic effects . . . . . . . . . . . . . . . . . . . . . . .
144 4.2.4 Relativistic dynamics . . . . . . . . . . . . . . . . . .
. . . 145 4.3 General theory of relativity and cosmology . . . . .
. . . . . . . . . . 148 4.3.1 Stars and galaxies . . . . . . . . .
. . . . . . . . . . . . . . 150 5 Mechanics of continuous media 153
5.1 Theory of elasticity . . . . . . . . . . . . . . . . . . . . .
. . . . . . 153 5.1.1 Stress . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 153 5.1.2 Elastic deformation . . . . . . . . .
. . . . . . . . . . . . . 156 5.1.3 Plastic deformation . . . . . .
. . . . . . . . . . . . . . . . 167 5.2 Hydrostatics, aerostatics .
. . . . . . . . . . . . . . . . . . . . . . . 171 5.2.1 Liquids and
gases . . . . . . . . . . . . . . . . . . . . . . . 172 5.2.2
Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
172 5.2.3 Buoyancy . . . . . . . . . . . . . . . . . . . . . . . .
. . . 180 5.2.4 Cohesion, adhesion, surface tension . . . . . . . .
. . . . . . 183 5.3 Hydrodynamics, aerodynamics . . . . . . . . . .
. . . . . . . . . . . 186 5.3.1 Flow eld . . . . . . . . . . . . .
. . . . . . . . . . . . . . 186 5.3.2 Basic equations of ideal ow .
. . . . . . . . . . . . . . . . 187 5.3.3 Real ow . . . . . . . . .
. . . . . . . . . . . . . . . . . . 197 5.3.4 Turbulent ow . . . .
. . . . . . . . . . . . . . . . . . . . . 203 5.3.5 Scaling laws .
. . . . . . . . . . . . . . . . . . . . . . . . . 206 5.3.6 Flow
with density variation . . . . . . . . . . . . . . . . . . 209 6
Nonlinear dynamics, chaos and fractals 211 6.1 Dynamical systems
and chaos . . . . . . . . . . . . . . . . . . . . . 212 6.1.1
Dynamical systems . . . . . . . . . . . . . . . . . . . . . . 212
6.1.2 Conservative systems . . . . . . . . . . . . . . . . . . . .
. 217 6.1.3 Dissipative systems . . . . . . . . . . . . . . . . . .
. . . . 219 6.2 Bifurcations . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 221 6.2.1 Logistic mapping . . . . . . . .
. . . . . . . . . . . . . . . 222 6.2.2 Universality . . . . . . .
. . . . . . . . . . . . . . . . . . . 225 6.3 Fractals . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Formula
symbols used in mechanics 229 7 Tables on mechanics 231 7.1 Density
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
231 7.1.1 Solids . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 231 7.1.2 Fluids . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 237 7.1.3 Gases . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 238
- 5. x Contents 7.2 Elastic properties . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 239 7.3 Dynamical properties . . . .
. . . . . . . . . . . . . . . . . . . . . . 243 7.3.1 Coefcients of
friction . . . . . . . . . . . . . . . . . . . . . 243 7.3.2
Compressibility . . . . . . . . . . . . . . . . . . . . . . . . 244
7.3.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 248 7.3.4 Flow resistance . . . . . . . . . . . . . . . . . . .
. . . . . 250 7.3.5 Surface tension . . . . . . . . . . . . . . . .
. . . . . . . . 251 Part II Vibrations and Waves 253 8 Vibrations
255 8.1 Free undamped vibrations . . . . . . . . . . . . . . . . .
. . . . . . 257 8.1.1 Mass on a spring . . . . . . . . . . . . . .
. . . . . . . . . . 258 8.1.2 Standard pendulum . . . . . . . . . .
. . . . . . . . . . . . 260 8.1.3 Physical pendulum . . . . . . . .
. . . . . . . . . . . . . . 263 8.1.4 Torsional vibration . . . . .
. . . . . . . . . . . . . . . . . 265 8.1.5 Liquid pendulum . . . .
. . . . . . . . . . . . . . . . . . . 266 8.1.6 Electric circuit .
. . . . . . . . . . . . . . . . . . . . . . . . 267 8.2 Damped
vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . .
268 8.2.1 Friction . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 269 8.2.2 Damped electric oscillator circuit . . . . . . .
. . . . . . . . 273 8.3 Forced vibrations . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 275 8.4 Superposition of vibrations .
. . . . . . . . . . . . . . . . . . . . . . 277 8.4.1 Superposition
of vibrations of equal frequency . . . . . . . . 277 8.4.2
Superposition of vibrations of different frequencies . . . . . .
279 8.4.3 Superposition of vibrations in different directions and
with different frequencies . . . . . . . . . . . 280 8.4.4 Fourier
analysis, decomposition into harmonics . . . . . . . . 282 8.5
Coupled vibrations . . . . . . . . . . . . . . . . . . . . . . . .
. . . 283 9 Waves 287 9.1 Basic features of waves . . . . . . . . .
. . . . . . . . . . . . . . . . 287 9.2 Polarization . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 293 9.3
Interference . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 294 9.3.1 Coherence . . . . . . . . . . . . . . . . . . . .
. . . . . . . 294 9.3.2 Interference . . . . . . . . . . . . . . .
. . . . . . . . . . . 295 9.3.3 Standing waves . . . . . . . . . .
. . . . . . . . . . . . . . 296 9.3.4 Waves with different
frequencies . . . . . . . . . . . . . . . 299 9.4 Doppler effect .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 9.4.1
Mach waves and Mach shock waves . . . . . . . . . . . . . . 302 9.5
Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 302 9.6 Reection . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 304 9.6.1 Phase relations . . . . . . . . . .
. . . . . . . . . . . . . . . 304 9.7 Dispersion . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 305 9.8 Diffraction . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 9.8.1
Diffraction by a slit . . . . . . . . . . . . . . . . . . . . . .
306 9.8.2 Diffraction by a grating . . . . . . . . . . . . . . . .
. . . . 307 9.9 Modulation of waves . . . . . . . . . . . . . . . .
. . . . . . . . . . 308 9.10 Surface waves and gravity waves . . .
. . . . . . . . . . . . . . . . . 309
- 6. Contents xi 10 Acoustics 311 10.1 Sound waves . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 311 10.1.1 Sound
velocity . . . . . . . . . . . . . . . . . . . . . . . . . 311
10.1.2 Parameters of sound . . . . . . . . . . . . . . . . . . . .
. . 313 10.1.3 Relative quantities . . . . . . . . . . . . . . . .
. . . . . . . 317 10.2 Sources and receivers of sound . . . . . . .
. . . . . . . . . . . . . . 319 10.2.1 Mechanical sound emitters .
. . . . . . . . . . . . . . . . . 319 10.2.2 Electro-acoustic
transducers . . . . . . . . . . . . . . . . . . 321 10.2.3 Sound
absorption . . . . . . . . . . . . . . . . . . . . . . . 324 10.2.4
Sound attenuation . . . . . . . . . . . . . . . . . . . . . . . 327
10.2.5 Flow noise . . . . . . . . . . . . . . . . . . . . . . . . .
. . 328 10.3 Ultrasound . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 328 10.4 Physiological acoustics and hearing . .
. . . . . . . . . . . . . . . . . 329 10.4.1 Perception of sound .
. . . . . . . . . . . . . . . . . . . . . 330 10.4.2 Evaluated
sound levels . . . . . . . . . . . . . . . . . . . . 331 10.5
Musical acoustics . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 331 11 Optics 335 11.1 Geometric optics . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 337 11.1.1 Optical
imagingfundamental concepts . . . . . . . . . . . 338 11.1.2
Reection . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
11.1.3 Refraction . . . . . . . . . . . . . . . . . . . . . . . . .
. . 345 11.2 Lenses . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 358 11.2.1 Thick lenses . . . . . . . . . . . . .
. . . . . . . . . . . . . 358 11.2.2 Thin lenses . . . . . . . . .
. . . . . . . . . . . . . . . . . 364 11.3 Lens systems . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 364 11.3.1 Lenses
with diaphragms . . . . . . . . . . . . . . . . . . . . 365 11.3.2
Image defects . . . . . . . . . . . . . . . . . . . . . . . . . 366
11.4 Optical instruments . . . . . . . . . . . . . . . . . . . . .
. . . . . . 368 11.4.1 Pinhole camera . . . . . . . . . . . . . . .
. . . . . . . . . 369 11.4.2 Camera . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 369 11.4.3 Eye . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 370 11.4.4 Eye and optical
instruments . . . . . . . . . . . . . . . . . . 372 11.5 Wave
optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 376 11.5.1 Scattering . . . . . . . . . . . . . . . . . . . . . .
. . . . . 376 11.5.2 Diffraction and limitation of resolution . . .
. . . . . . . . . 377 11.5.3 Refraction in the wave picture . . . .
. . . . . . . . . . . . . 379 11.5.4 Interference . . . . . . . . .
. . . . . . . . . . . . . . . . . 380 11.5.5 Diffractive optical
elements . . . . . . . . . . . . . . . . . . 384 11.5.6 Dispersion
. . . . . . . . . . . . . . . . . . . . . . . . . . . 389 11.5.7
Spectroscopic apparatus . . . . . . . . . . . . . . . . . . . . 390
11.5.8 Polarization of light . . . . . . . . . . . . . . . . . . .
. . . 391 11.6 Photometry . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 395 11.6.1 Photometric quantities . . . . . . .
. . . . . . . . . . . . . . 396 11.6.2 Photometric quantities . . .
. . . . . . . . . . . . . . . . . . 403 Symbols used in formulae on
vibrations, waves, acoustics and optics 407 12 Tables on
vibrations, waves, acoustics and optics 409 12.1 Tables on
vibrations and acoustics . . . . . . . . . . . . . . . . . . . 409
12.2 Tables on optics . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 414
- 7. xii Contents Part III Electricity 419 13 Charges and
currents 421 13.1 Electric charge . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 421 13.1.1 Coulombs law . . . . . . . . . .
. . . . . . . . . . . . . . 423 13.2 Electric charge density . . .
. . . . . . . . . . . . . . . . . . . . . . 424 13.3 Electric
current . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
426 13.3.1 Amperes law . . . . . . . . . . . . . . . . . . . . . .
. . . 428 13.4 Electric current density . . . . . . . . . . . . . .
. . . . . . . . . . . 428 13.4.1 Electric current ow eld . . . . .
. . . . . . . . . . . . . . 430 13.5 Electric resistance and
conductance . . . . . . . . . . . . . . . . . . . 431 13.5.1
Electric resistance . . . . . . . . . . . . . . . . . . . . . . .
431 13.5.2 Electric conductance . . . . . . . . . . . . . . . . . .
. . . . 432 13.5.3 Resistivity and conductivity . . . . . . . . . .
. . . . . . . . 432 13.5.4 Mobility of charge carriers . . . . . .
. . . . . . . . . . . . 433 13.5.5 Temperature dependence of the
resistance . . . . . . . . . . . 434 13.5.6 Variable resistors . .
. . . . . . . . . . . . . . . . . . . . . 435 13.5.7 Connection of
resistors . . . . . . . . . . . . . . . . . . . . 436 14 Electric
and magnetic elds 439 14.1 Electric eld . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 439 14.2 Electrostatic induction .
. . . . . . . . . . . . . . . . . . . . . . . . 440 14.2.1 Electric
eld lines . . . . . . . . . . . . . . . . . . . . . . . 441 14.2.2
Electric eld strength of point charges . . . . . . . . . . . . .
444 14.3 Force . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 445 14.4 Electric voltage . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 445 14.5 Electric potential . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 447 14.5.1
Equipotential surfaces . . . . . . . . . . . . . . . . . . . . .
448 14.5.2 Field strength and potential of various charge
distributions . . 448 14.5.3 Electric ux . . . . . . . . . . . . .
. . . . . . . . . . . . . 451 14.5.4 Electric displacement in a
vacuum . . . . . . . . . . . . . . . 453 14.6 Electric polarization
. . . . . . . . . . . . . . . . . . . . . . . . . . 454 14.6.1
Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . .
456 14.7 Capacitance . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 457 14.7.1 Parallel-plate capacitor . . . . . . . .
. . . . . . . . . . . . 458 14.7.2 Parallel connection of
capacitors . . . . . . . . . . . . . . . 458 14.7.3 Series
connection of capacitors . . . . . . . . . . . . . . . . 459 14.7.4
Capacitance of simple arrangements of conductors . . . . . . 459
14.8 Energy and energy density of the electric eld . . . . . . . .
. . . . . 460 14.9 Electric eld at interfaces . . . . . . . . . . .
. . . . . . . . . . . . . 461 14.10 Magnetic eld . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 462 14.11 Magnetism . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 463 14.11.1
Magnetic eld lines . . . . . . . . . . . . . . . . . . . . . . 463
14.12 Magnetic ux density . . . . . . . . . . . . . . . . . . . . .
. . . . . 465 14.13 Magnetic ux . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 467 14.14 Magnetic eld strength . . . . . .
. . . . . . . . . . . . . . . . . . . 469 14.15 Magnetic potential
difference and magnetic circuits . . . . . . . . . . 470 14.15.1
Amperes law . . . . . . . . . . . . . . . . . . . . . . . . . 472
14.15.2 Biot-Savarts law . . . . . . . . . . . . . . . . . . . . .
. . 474 14.15.3 Magnetic eld of a rectilinear conductor . . . . . .
. . . . . 476 14.15.4 Magnetic elds of various current
distributions . . . . . . . . 477
- 8. Contents xiii 14.16 Matter in magnetic elds . . . . . . . .
. . . . . . . . . . . . . . . . 478 14.16.1 Diamagnetism . . . . .
. . . . . . . . . . . . . . . . . . . . 480 14.16.2 Paramagnetism .
. . . . . . . . . . . . . . . . . . . . . . . 480 14.16.3
Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . 481
14.16.4 Antiferromagnetism . . . . . . . . . . . . . . . . . . . .
. . 483 14.16.5 Ferrimagnetism . . . . . . . . . . . . . . . . . .
. . . . . . 484 14.17 Magnetic elds at interfaces . . . . . . . . .
. . . . . . . . . . . . . 484 14.18 Induction . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 485 14.18.1 Faradays law
of induction . . . . . . . . . . . . . . . . . . . 486 14.18.2
Transformer induction . . . . . . . . . . . . . . . . . . . . . 487
14.19 Self-induction . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 488 14.19.1 Inductances of geometric arrangements of
conductors . . . . . 490 14.19.2 Magnetic conductance . . . . . . .
. . . . . . . . . . . . . . 491 14.20 Mutual induction . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 492 14.20.1 Transformer
. . . . . . . . . . . . . . . . . . . . . . . . . . 493 14.21
Energy and energy density of the magnetic eld . . . . . . . . . . .
. 494 14.22 Maxwells equations . . . . . . . . . . . . . . . . . .
. . . . . . . . 496 14.22.1 Displacement current . . . . . . . . .
. . . . . . . . . . . . 497 14.22.2 Electromagnetic waves . . . . .
. . . . . . . . . . . . . . . 498 14.22.3 Poynting vector . . . . .
. . . . . . . . . . . . . . . . . . . 500 15 Applications in
electrical engineering 501 15.1 Direct-current circuit . . . . . .
. . . . . . . . . . . . . . . . . . . . 502 15.1.1 Kirchhoffs laws
for direct-current circuit . . . . . . . . . . . 503 15.1.2
Resistors in a direct-current circuit . . . . . . . . . . . . . .
503 15.1.3 Real voltage source . . . . . . . . . . . . . . . . . .
. . . . 505 15.1.4 Power and energy in the direct-current circuit .
. . . . . . . . 507 15.1.5 Matching for power transfer . . . . . .
. . . . . . . . . . . . 508 15.1.6 Measurement of current and
voltage . . . . . . . . . . . . . . 509 15.1.7 Resistance
measurement by means of the compensation method . . . . . . . . . .
. . . . . . . . . . . 510 15.1.8 Charging and discharging of
capacitors . . . . . . . . . . . . 511 15.1.9 Switching the current
on and off in a RL-circuit . . . . . . . 513 15.2
Alternating-current circuit . . . . . . . . . . . . . . . . . . . .
. . . 514 15.2.1 Alternating quantities . . . . . . . . . . . . . .
. . . . . . . 514 15.2.2 Representation of sinusoidal quantities in
a phasor diagram . . 517 15.2.3 Calculation rules for phasor
quantities . . . . . . . . . . . . . 519 15.2.4 Basics of
alternating-current engineering . . . . . . . . . . . 522 15.2.5
Basic components in the alternating-current circuit . . . . . . 529
15.2.6 Series connection of resistor and capacitor . . . . . . . .
. . 534 15.2.7 Parallel connection of a resistor and a capacitor .
. . . . . . . 535 15.2.8 Parallel connection of a resistor and an
inductor . . . . . . . . 536 15.2.9 Series connection of a resistor
and an inductor . . . . . . . . 536 15.2.10 Series-resonant circuit
. . . . . . . . . . . . . . . . . . . . . 538 15.2.11
Parallel-resonant circuit . . . . . . . . . . . . . . . . . . . .
539 15.2.12 Equivalence of series and parallel connections . . . .
. . . . 541 15.2.13 Radio waves . . . . . . . . . . . . . . . . . .
. . . . . . . . 542 15.3 Electric machines . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 544 15.3.1 Fundamental functional
principle . . . . . . . . . . . . . . . 544 15.3.2 Direct-current
machine . . . . . . . . . . . . . . . . . . . . 545 15.3.3
Three-phase machine . . . . . . . . . . . . . . . . . . . . .
547
- 9. xiv Contents 16 Current conduction in liquids, gases and
vacuum 551 16.1 Electrolysis . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 551 16.1.1 Amount of substance . . . . . .
. . . . . . . . . . . . . . . 551 16.1.2 Ions . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 552 16.1.3 Electrodes . . . .
. . . . . . . . . . . . . . . . . . . . . . . 552 16.1.4
Electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . .
552 16.1.5 Galvanic cells . . . . . . . . . . . . . . . . . . . . .
. . . . 557 16.1.6 Electrokinetic effects . . . . . . . . . . . . .
. . . . . . . . 560 16.2 Current conduction in gases . . . . . . .
. . . . . . . . . . . . . . . . 560 16.2.1 Non-self-sustained
discharge . . . . . . . . . . . . . . . . . 560 16.2.2
Self-sustained gaseous discharge . . . . . . . . . . . . . . . 563
16.3 Electron emission . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 565 16.3.1 Thermo-ionic emission . . . . . . . . . . .
. . . . . . . . . 565 16.3.2 Photo emission . . . . . . . . . . . .
. . . . . . . . . . . . 565 16.3.3 Field emission . . . . . . . . .
. . . . . . . . . . . . . . . . 566 16.3.4 Secondary electron
emission . . . . . . . . . . . . . . . . . 567 16.4 Vacuum tubes .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
16.4.1 Vacuum-tube diode . . . . . . . . . . . . . . . . . . . . .
. 568 16.4.2 Vacuum-tube triode . . . . . . . . . . . . . . . . . .
. . . . 568 16.4.3 Tetrode . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 571 16.4.4 Cathode rays . . . . . . . . . . . . . .
. . . . . . . . . . . . 571 16.4.5 Channel rays . . . . . . . . . .
. . . . . . . . . . . . . . . . 571 17 Plasma physics 573 17.1
Properties of a plasma . . . . . . . . . . . . . . . . . . . . . .
. . . 573 17.1.1 Plasma parameters . . . . . . . . . . . . . . . .
. . . . . . . 573 17.1.2 Plasma radiation . . . . . . . . . . . . .
. . . . . . . . . . . 580 17.1.3 Plasmas in magnetic elds . . . . .
. . . . . . . . . . . . . . 581 17.1.4 Plasma waves . . . . . . . .
. . . . . . . . . . . . . . . . . 583 17.2 Generation of plasmas .
. . . . . . . . . . . . . . . . . . . . . . . . 586 17.2.1 Thermal
generation of plasma . . . . . . . . . . . . . . . . . 586 17.2.2
Generation of plasma by compression . . . . . . . . . . . . . 586
17.3 Energy production with plasmas . . . . . . . . . . . . . . . .
. . . . 588 17.3.1 MHD generator . . . . . . . . . . . . . . . . .
. . . . . . . 588 17.3.2 Nuclear fusion reactors . . . . . . . . .
. . . . . . . . . . . 589 17.3.3 Fusion with magnetic connement . .
. . . . . . . . . . . . 590 17.3.4 Fusion with inertial connement .
. . . . . . . . . . . . . . 591 Symbols used in formulae on
electricity and plasma physics 593 18 Tables on electricity 595
18.1 Metals and alloys . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 595 18.1.1 Specic electric resistance . . . . . . . . .
. . . . . . . . . . 595 18.1.2 Electrochemical potential series . .
. . . . . . . . . . . . . . 598 18.2 Dielectrics . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 601 18.3 Practical
tables of electric engineering . . . . . . . . . . . . . . . . .
606 18.4 Magnetic properties . . . . . . . . . . . . . . . . . . .
. . . . . . . . 609 18.5 Ferromagnetic properties . . . . . . . . .
. . . . . . . . . . . . . . . 614 18.5.1 Magnetic anisotropy . . .
. . . . . . . . . . . . . . . . . . . 617 18.6 Ferrites . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 619 18.7
Antiferromagnets . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 619 18.8 Ion mobility . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 620
- 10. Contents xv Part IV Thermodynamics 621 19 Equilibrium and
state variables 623 19.1 Systems, phases and equilibrium . . . . .
. . . . . . . . . . . . . . . 623 19.1.1 Systems . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 623 19.1.2 Phases . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 624 19.1.3
Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 625
19.2 State variables . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 627 19.2.1 State property denitions . . . . . . . . . .
. . . . . . . . . 627 19.2.2 Temperature . . . . . . . . . . . . .
. . . . . . . . . . . . . 629 19.2.3 Pressure . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 634 19.2.4 Particle number,
amount of substance and Avogadro number . 637 19.2.5 Entropy . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 640 19.3
Thermodynamic potentials . . . . . . . . . . . . . . . . . . . . .
. . 641 19.3.1 Principle of maximum entropyprinciple of minimum
energy . . . . . . . . . . . . . . . . . . . . . . . 641 19.3.2
Internal energy as a potential . . . . . . . . . . . . . . . . .
641 19.3.3 Entropy as a thermodynamic potential . . . . . . . . . .
. . . 642 19.3.4 Free energy . . . . . . . . . . . . . . . . . . .
. . . . . . . 643 19.3.5 Enthalpy . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 644 19.3.6 Free enthalpy . . . . . . . . . .
. . . . . . . . . . . . . . . 647 19.3.7 Maxwell relations . . . .
. . . . . . . . . . . . . . . . . . . 648 19.3.8 Thermodynamic
stability . . . . . . . . . . . . . . . . . . . 649 19.4 Ideal gas
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
19.4.1 Boyle-Mariotte law . . . . . . . . . . . . . . . . . . . . .
. 651 19.4.2 Law of Gay-Lussac . . . . . . . . . . . . . . . . . .
. . . . 651 19.4.3 Equation of state . . . . . . . . . . . . . . .
. . . . . . . . . 652 19.5 Kinetic theory of the ideal gas . . . .
. . . . . . . . . . . . . . . . . 653 19.5.1 Pressure and
temperature . . . . . . . . . . . . . . . . . . . 653 19.5.2
MaxwellBoltzmann distribution . . . . . . . . . . . . . . . 655
19.5.3 Degrees of freedom . . . . . . . . . . . . . . . . . . . . .
. 657 19.5.4 Equipartition law . . . . . . . . . . . . . . . . . .
. . . . . 657 19.5.5 Transport processes . . . . . . . . . . . . .
. . . . . . . . . 658 19.6 Equations of state . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 661 19.6.1 Equation of state of
the ideal gas . . . . . . . . . . . . . . . 661 19.6.2 Equation of
state of real gases . . . . . . . . . . . . . . . . . 665 19.6.3
Equation of states for liquids and solids . . . . . . . . . . . .
671 20 Heat, conversion of energy and changes of state 675 20.1
Energy forms . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 675 20.1.1 Energy units . . . . . . . . . . . . . . . . . . .
. . . . . . . 675 20.1.2 Work . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 676 20.1.3 Chemical potential . . . . . . . . .
. . . . . . . . . . . . . . 677 20.1.4 Heat . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 678 20.2 Energy conversion . .
. . . . . . . . . . . . . . . . . . . . . . . . . 679 20.2.1
Conversion of equivalent energies into heat . . . . . . . . . . 679
20.2.2 Conversion of heat into other forms of energy . . . . . . .
. . 683 20.2.3 Exergy and anergy . . . . . . . . . . . . . . . . .
. . . . . . 683 20.3 Heat capacity . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 684 20.3.1 Total heat capacity . . . .
. . . . . . . . . . . . . . . . . . . 684 20.3.2 Molar heat
capacity . . . . . . . . . . . . . . . . . . . . . . 686 20.3.3
Specic heat capacity . . . . . . . . . . . . . . . . . . . . .
687
- 11. xvi Contents 20.4 Changes of state . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 691 20.4.1 Reversible and
irreversible processes . . . . . . . . . . . . . 691 20.4.2
Isothermal processes . . . . . . . . . . . . . . . . . . . . . .
692 20.4.3 Isobaric processes . . . . . . . . . . . . . . . . . . .
. . . . 693 20.4.4 Isochoric processes . . . . . . . . . . . . . .
. . . . . . . . 694 20.4.5 Adiabatic (isentropic) processes . . . .
. . . . . . . . . . . . 695 20.4.6 Equilibrium states . . . . . . .
. . . . . . . . . . . . . . . . 697 20.5 Laws of thermodynamics . .
. . . . . . . . . . . . . . . . . . . . . . 698 20.5.1 Zeroth law
of thermodynamics . . . . . . . . . . . . . . . . 698 20.5.2 First
law of thermodynamics . . . . . . . . . . . . . . . . . 698 20.5.3
Second law of thermodynamics . . . . . . . . . . . . . . . . 701
20.5.4 Third law of thermodynamics . . . . . . . . . . . . . . . .
. 702 20.6 Carnot cycle . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 702 20.6.1 Principle and application . . . . . . .
. . . . . . . . . . . . 702 20.6.2 Reduced heat . . . . . . . . . .
. . . . . . . . . . . . . . . 705 20.7 Thermodynamic machines . . .
. . . . . . . . . . . . . . . . . . . . 706 20.7.1 Right-handed and
left-handed processes . . . . . . . . . . . . 706 20.7.2 Heat pump
and refrigerator . . . . . . . . . . . . . . . . . . 707 20.7.3
Stirling cycle . . . . . . . . . . . . . . . . . . . . . . . . .
708 20.7.4 Steam engine . . . . . . . . . . . . . . . . . . . . . .
. . . 709 20.7.5 Open systems . . . . . . . . . . . . . . . . . . .
. . . . . . 710 20.7.6 Otto and Diesel engines . . . . . . . . . .
. . . . . . . . . . 711 20.7.7 Gas turbines . . . . . . . . . . . .
. . . . . . . . . . . . . . 713 20.8 Gas liquefaction . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 714 20.8.1 Generation of
low temperatures . . . . . . . . . . . . . . . . 714 20.8.2
JouleThomson effect . . . . . . . . . . . . . . . . . . . . . 715
21 Phase transitions, reactions and equalizing of heat 717 21.1
Phase and state of aggregation . . . . . . . . . . . . . . . . . .
. . . 717 21.1.1 Phase . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 717 21.1.2 Aggregation states . . . . . . . . . . . .
. . . . . . . . . . . 717 21.1.3 Conversions of aggregation states
. . . . . . . . . . . . . . . 718 21.1.4 Vapor . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 719 21.2 Order of phase
transitions . . . . . . . . . . . . . . . . . . . . . . . . 720
21.2.1 First-order phase transition . . . . . . . . . . . . . . . .
. . 720 21.2.2 Second-order phase transition . . . . . . . . . . .
. . . . . . 721 21.2.3 Lambda transitions . . . . . . . . . . . . .
. . . . . . . . . 722 21.2.4 Phase-coexistence region . . . . . . .
. . . . . . . . . . . . 722 21.2.5 Critical indices . . . . . . . .
. . . . . . . . . . . . . . . . 723 21.3 Phase transition and Van
der Waals gas . . . . . . . . . . . . . . . . . 724 21.3.1 Phase
equilibrium . . . . . . . . . . . . . . . . . . . . . . . 724
21.3.2 Maxwell construction . . . . . . . . . . . . . . . . . . . .
. 724 21.3.3 Delayed boiling and delayed condensation . . . . . . .
. . . 726 21.3.4 Theorem of corresponding states . . . . . . . . .
. . . . . . 727 21.4 Examples of phase transitions . . . . . . . .
. . . . . . . . . . . . . . 727 21.4.1 Magnetic phase transitions .
. . . . . . . . . . . . . . . . . 727 21.4.2 Orderdisorder phase
transitions . . . . . . . . . . . . . . . 728 21.4.3 Change in the
crystal structure . . . . . . . . . . . . . . . . . 729 21.4.4
Liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . .
730 21.4.5 Superconductivity . . . . . . . . . . . . . . . . . . .
. . . . 730 21.4.6 Superuidity . . . . . . . . . . . . . . . . . .
. . . . . . . . 731
- 12. Contents xvii 21.5 Multicomponent gases . . . . . . . . . .
. . . . . . . . . . . . . . . 731 21.5.1 Partial pressure and
Daltons law . . . . . . . . . . . . . . . 732 21.5.2 Euler equation
and GibbsDuhem relation . . . . . . . . . . 733 21.6 Multiphase
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
21.6.1 Phase equilibrium . . . . . . . . . . . . . . . . . . . . .
. . 734 21.6.2 Gibbs phase rule . . . . . . . . . . . . . . . . . .
. . . . . . 734 21.6.3 ClausiusClapeyron equation . . . . . . . . .
. . . . . . . . 735 21.7 Vapor pressure of solutions . . . . . . .
. . . . . . . . . . . . . . . . 736 21.7.1 Raoults law . . . . . .
. . . . . . . . . . . . . . . . . . . . 736 21.7.2 Boiling-point
elevation and freezing-point depression . . . . . 736 21.7.3
HenryDalton law . . . . . . . . . . . . . . . . . . . . . . . 738
21.7.4 Steamair mixtures (humid air) . . . . . . . . . . . . . . .
. 738 21.8 Chemical reactions . . . . . . . . . . . . . . . . . . .
. . . . . . . . 742 21.8.1 Stoichiometry . . . . . . . . . . . . .
. . . . . . . . . . . . 743 21.8.2 Phase rule for chemical
reactions . . . . . . . . . . . . . . . 744 21.8.3 Law of mass
action . . . . . . . . . . . . . . . . . . . . . . 744 21.8.4
pH-value and solubility product . . . . . . . . . . . . . . . . 746
21.9 Equalization of temperature . . . . . . . . . . . . . . . . .
. . . . . . 748 21.9.1 Mixing temperature of two systems . . . . .
. . . . . . . . . 748 21.9.2 Reversible and irreversible processes
. . . . . . . . . . . . . 749 21.10 Heat transfer . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 750 21.10.1 Heat ow . . .
. . . . . . . . . . . . . . . . . . . . . . . . 751 21.10.2 Heat
transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 751
21.10.3 Heat conduction . . . . . . . . . . . . . . . . . . . . . .
. . 753 21.10.4 Thermal resistance . . . . . . . . . . . . . . . .
. . . . . . . 757 21.10.5 Heat transmission . . . . . . . . . . . .
. . . . . . . . . . . 759 21.10.6 Heat radiation . . . . . . . . .
. . . . . . . . . . . . . . . . 764 21.10.7 Deposition of radiation
. . . . . . . . . . . . . . . . . . . . 764 21.11 Transport of heat
and mass . . . . . . . . . . . . . . . . . . . . . . . 766 21.11.1
Fouriers law . . . . . . . . . . . . . . . . . . . . . . . . . .
766 21.11.2 Continuity equation . . . . . . . . . . . . . . . . . .
. . . . 766 21.11.3 Heat conduction equation . . . . . . . . . . .
. . . . . . . . 767 21.11.4 Ficks law and diffusion equation . . .
. . . . . . . . . . . . 768 21.11.5 Solution of the equation of
heat conduction and diffusion . . . 769 Formula symbols used in
thermodynamics 771 22 Tables on thermodynamics 775 22.1
Characteristic temperatures . . . . . . . . . . . . . . . . . . . .
. . . 775 22.1.1 Units and calibration points . . . . . . . . . . .
. . . . . . . 775 22.1.2 Melting and boiling points . . . . . . . .
. . . . . . . . . . . 777 22.1.3 Curie and Neel temperatures . . .
. . . . . . . . . . . . . . . 786 22.2 Characteristics of real
gases . . . . . . . . . . . . . . . . . . . . . . . 787 22.3
Thermal properties of substances . . . . . . . . . . . . . . . . .
. . . 788 22.3.1 Viscosity . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 788 22.3.2 Expansion, heat capacity and thermal
conductivity . . . . . . 789 22.4 Heat transmission . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 795 22.5 Practical
correction data . . . . . . . . . . . . . . . . . . . . . . . . 798
22.5.1 Pressure measurement . . . . . . . . . . . . . . . . . . . .
. 798 22.5.2 Volume measurementsconversion to standard temperature
. 803 22.6 Generation of liquid low-temperature baths . . . . . . .
. . . . . . . . 804
- 13. xviii Contents 22.7 Dehydrators . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 805 22.8 Vapor pressure . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 806 22.8.1
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 806
22.8.2 Relative humidity . . . . . . . . . . . . . . . . . . . . .
. . 806 22.8.3 Vapor pressure of water . . . . . . . . . . . . . .
. . . . . . 807 22.9 Specic enthalpies . . . . . . . . . . . . . .
. . . . . . . . . . . . . 809 Part V Quantum physics 815 23
Photons, electromagnetic radiation and light quanta 817 23.1
Plancks radiation law . . . . . . . . . . . . . . . . . . . . . . .
. . . 817 23.2 Photoelectric effect . . . . . . . . . . . . . . . .
. . . . . . . . . . . 820 23.3 Compton effect . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 822 24 Matter waveswave
mechanics of particles 825 24.1 Wave character of particles . . . .
. . . . . . . . . . . . . . . . . . . 825 24.2 Heisenbergs
uncertainty principle . . . . . . . . . . . . . . . . . . . 827
24.3 Wave function and observable . . . . . . . . . . . . . . . . .
. . . . 827 24.4 Schrodinger equation . . . . . . . . . . . . . . .
. . . . . . . . . . . 835 24.4.1 Piecewise constant potentials . .
. . . . . . . . . . . . . . . 837 24.4.2 Harmonic oscillator . . .
. . . . . . . . . . . . . . . . . . . 841 24.4.3 Pauli principle .
. . . . . . . . . . . . . . . . . . . . . . . . 843 24.5 Spin and
magnetic moments . . . . . . . . . . . . . . . . . . . . . . 844
24.5.1 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 844 24.5.2 Magnetic moments . . . . . . . . . . . . . . . . . .
. . . . 847 25 Atomic and molecular physics 851 25.1 Fundamentals
of spectroscopy . . . . . . . . . . . . . . . . . . . . . 851 25.2
Hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 854 25.2.1 Bohrs postulates . . . . . . . . . . . . . . . . . .
. . . . . 855 25.3 Stationary states and quantum numbers in the
central eld . . . . . . . 859 25.4 Many-electron atoms . . . . . .
. . . . . . . . . . . . . . . . . . . . 864 25.5 X-rays . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 868 25.5.1
Applications of x-rays . . . . . . . . . . . . . . . . . . . . .
870 25.6 Molecular spectra . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 871 25.7 Atoms in external elds . . . . . . . . . .
. . . . . . . . . . . . . . . 874 25.8 Periodic Table of elements .
. . . . . . . . . . . . . . . . . . . . . . 877 25.9 Interaction of
photons with atoms and molecules . . . . . . . . . . . . 879 25.9.1
Spontaneous and induced emission . . . . . . . . . . . . . . 879 26
Elementary particle physicsstandard model 883 26.1 Unication of
interactions . . . . . . . . . . . . . . . . . . . . . . . 883
26.1.1 Standard model . . . . . . . . . . . . . . . . . . . . . . .
. 883 26.1.2 Field quanta or gauge bosons . . . . . . . . . . . . .
. . . . 887 26.1.3 Fermions and bosons . . . . . . . . . . . . . .
. . . . . . . 889 26.2 Leptons, quarks, and vector bosons . . . . .
. . . . . . . . . . . . . . 891 26.2.1 Leptons . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 891 26.2.2 Quarks . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 892 26.2.3 Hadrons . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 894 26.2.4
Accelerators and detectors . . . . . . . . . . . . . . . . . . .
898
- 14. Contents xix 26.3 Symmetries and conservation laws . . . .
. . . . . . . . . . . . . . . 900 26.3.1 Parity conservation and
the weak interaction . . . . . . . . . 900 26.3.2 Charge
conservation and pair production . . . . . . . . . . . 901 26.3.3
Charge conjugation and antiparticles . . . . . . . . . . . . . 902
26.3.4 Time-reversal invariance and inverse reactions . . . . . . .
. 903 26.3.5 Conservation laws . . . . . . . . . . . . . . . . . .
. . . . . 903 26.3.6 Beyond the standard model . . . . . . . . . .
. . . . . . . . 904 27 Nuclear physics 907 27.1 Constituents of the
atomic nucleus . . . . . . . . . . . . . . . . . . . 907 27.2 Basic
quantities of the atomic nucleus . . . . . . . . . . . . . . . . .
910 27.3 Nucleon-nucleon interaction . . . . . . . . . . . . . . .
. . . . . . . 912 27.3.1 Phenomenologic nucleon-nucleon potentials
. . . . . . . . . 912 27.3.2 Meson exchange potentials . . . . . .
. . . . . . . . . . . . 914 27.4 Nuclear models . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 915 27.4.1 Fermi-gas model .
. . . . . . . . . . . . . . . . . . . . . . . 915 27.4.2 Nuclear
matter . . . . . . . . . . . . . . . . . . . . . . . . . 915 27.4.3
Droplet model . . . . . . . . . . . . . . . . . . . . . . . . . 916
27.4.4 Shell model . . . . . . . . . . . . . . . . . . . . . . . .
. . 917 27.4.5 Collective model . . . . . . . . . . . . . . . . . .
. . . . . . 920 27.5 Nuclear reactions . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 922 27.5.1 Reaction channels and
cross-sections . . . . . . . . . . . . . 922 27.5.2 Conservation
laws in nuclear reactions . . . . . . . . . . . . 926 27.5.3
Elastic scattering . . . . . . . . . . . . . . . . . . . . . . .
928 27.5.4 Compound-nuclear reactions . . . . . . . . . . . . . . .
. . 929 27.5.5 Optical model . . . . . . . . . . . . . . . . . . .
. . . . . . 931 27.5.6 Direct reactions . . . . . . . . . . . . . .
. . . . . . . . . . 931 27.5.7 Heavy-ion reactions . . . . . . . .
. . . . . . . . . . . . . . 932 27.5.8 Nuclear ssion . . . . . . .
. . . . . . . . . . . . . . . . . . 935 27.6 Nuclear decay . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 937 27.6.1
Decay law . . . . . . . . . . . . . . . . . . . . . . . . . . . 938
27.6.2 -decay . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 941 27.6.3 -decay . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 943 27.6.4 -decay . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 946 27.6.5 Emission of nucleons and nucleon clusters
. . . . . . . . . . 947 27.7 Nuclear reactor . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 947 27.7.1 Types of reactors .
. . . . . . . . . . . . . . . . . . . . . . 949 27.8 Nuclear fusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 950 27.9
Interaction of radiation with matter . . . . . . . . . . . . . . .
. . . . 953 27.9.1 Ionizing particles . . . . . . . . . . . . . . .
. . . . . . . . 953 27.9.2 -radiation . . . . . . . . . . . . . . .
. . . . . . . . . . . 956 27.10 Dosimetry . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 958 27.10.1 Methods of dosage
measurements . . . . . . . . . . . . . . . 962 27.10.2
Environmental radioactivity . . . . . . . . . . . . . . . . . . 964
28 Solid-state physics 967 28.1 Structure of solid bodies . . . . .
. . . . . . . . . . . . . . . . . . . 967 28.1.1 Basic concepts of
solid-state physics . . . . . . . . . . . . . 967 28.1.2 Structure
of crystals . . . . . . . . . . . . . . . . . . . . . . 968 28.1.3
Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . .
970 28.1.4 Methods for structure investigation . . . . . . . . . .
. . . . 974 28.1.5 Bond relations in crystals . . . . . . . . . . .
. . . . . . . . 976
- 15. xx Contents 28.2 Lattice defects . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 979 28.2.1 Point defects . . . . .
. . . . . . . . . . . . . . . . . . . . . 979 28.2.2
One-dimensional defects . . . . . . . . . . . . . . . . . . . 981
28.2.3 Two-dimensional lattice defects . . . . . . . . . . . . . .
. . 982 28.2.4 Amorphous solids . . . . . . . . . . . . . . . . . .
. . . . . 983 28.3 Mechanical properties of materials . . . . . . .
. . . . . . . . . . . . 984 28.3.1 Macromolecular solids . . . . .
. . . . . . . . . . . . . . . 984 28.3.2 Compound materials . . . .
. . . . . . . . . . . . . . . . . . 987 28.3.3 Alloys . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 988 28.3.4 Liquid
crystals . . . . . . . . . . . . . . . . . . . . . . . . . 990 28.4
Phonons and lattice vibrations . . . . . . . . . . . . . . . . . .
. . . 991 28.4.1 Elastic waves . . . . . . . . . . . . . . . . . .
. . . . . . . 991 28.4.2 Phonons and specic heat capacity . . . . .
. . . . . . . . . 995 28.4.3 Einstein model . . . . . . . . . . . .
. . . . . . . . . . . . . 996 28.4.4 Debye model . . . . . . . . .
. . . . . . . . . . . . . . . . 997 28.4.5 Heat conduction . . . .
. . . . . . . . . . . . . . . . . . . . 999 28.5 Electrons in
solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000
28.5.1 Free-electron gas . . . . . . . . . . . . . . . . . . . . .
. . 1001 28.5.2 Band model . . . . . . . . . . . . . . . . . . . .
. . . . . . 1007 28.6 Semiconductors . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 1011 28.6.1 Extrinsic conduction . . .
. . . . . . . . . . . . . . . . . . . 1014 28.6.2 Semiconductor
diode . . . . . . . . . . . . . . . . . . . . . 1016 28.6.3
Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . .
1023 28.6.4 Unipolar (eld effect) transistors . . . . . . . . . . .
. . . . 1030 28.6.5 Thyristor . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 1032 28.6.6 Integrated circuits (IC) . . . . . .
. . . . . . . . . . . . . . . 1034 28.6.7 Operational ampliers . .
. . . . . . . . . . . . . . . . . . . 1037 28.7 Superconductivity .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1042 28.7.1
Fundamental properties of superconductivity . . . . . . . . . 1043
28.7.2 High-temperature superconductors . . . . . . . . . . . . . .
1047 28.8 Magnetic properties . . . . . . . . . . . . . . . . . . .
. . . . . . . . 1049 28.8.1 Ferromagnetism . . . . . . . . . . . .
. . . . . . . . . . . . 1052 28.8.2 Antiferromagnetism and
ferrimagnetism . . . . . . . . . . . 1054 28.9 Dielectric
properties . . . . . . . . . . . . . . . . . . . . . . . . . . 1055
28.9.1 Para-electric materials . . . . . . . . . . . . . . . . . .
. . . 1059 28.9.2 Ferroelectrics . . . . . . . . . . . . . . . . .
. . . . . . . . 1059 28.10 Optical properties of crystals . . . . .
. . . . . . . . . . . . . . . . . 1060 28.10.1 Excitons and their
properties . . . . . . . . . . . . . . . . . 1061 28.10.2
Photoconductivity . . . . . . . . . . . . . . . . . . . . . . .
1062 28.10.3 Luminescence . . . . . . . . . . . . . . . . . . . . .
. . . . 1063 28.10.4 Optoelectronic properties . . . . . . . . . .
. . . . . . . . . 1063 Formula symbols used in quantum physics 1065
29 Tables in quantum physics 1071 29.1 Ionization potentials . . .
. . . . . . . . . . . . . . . . . . . . . . . 1071 29.2 Atomic and
ionic radii of elements . . . . . . . . . . . . . . . . . . . 1078
29.3 Electron emission . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 1082 29.4 X-rays . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 1086 29.5 Nuclear reactions . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 1087 29.6 Interaction
of radiation with matter . . . . . . . . . . . . . . . . . . .
1088
- 16. Contents xxi 29.7 Hall effect . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 1089 29.8 Superconductors . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 1091 29.9
Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1093 29.9.1 Thermal, magnetic and electric properties of
semiconductors . 1093 Part VI Appendix 1095 30 Measurements and
measurement errors 1097 30.1 Description of measurements . . . . .
. . . . . . . . . . . . . . . . . 1097 30.1.1 Quantities and SI
units . . . . . . . . . . . . . . . . . . . . . 1097 30.2 Error
theory and statistics . . . . . . . . . . . . . . . . . . . . . . .
. 1100 30.2.1 Types of errors . . . . . . . . . . . . . . . . . . .
. . . . . . 1100 30.2.2 Mean values of runs . . . . . . . . . . . .
. . . . . . . . . . 1102 30.2.3 Variance . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 1104 30.2.4 Correlation . . . . . . .
. . . . . . . . . . . . . . . . . . . 1105 30.2.5 Regression
analysis . . . . . . . . . . . . . . . . . . . . . . 1106 30.2.6
Rate distributions . . . . . . . . . . . . . . . . . . . . . . .
1106 30.2.7 Reliability . . . . . . . . . . . . . . . . . . . . . .
. . . . . 1111 31 Vector calculus 1115 31.1.1 Vectors . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 1115 31.1.2
Multiplication by a scalar . . . . . . . . . . . . . . . . . . .
1116 31.1.3 Addition and subtraction of vectors . . . . . . . . . .
. . . . 1117 31.1.4 Multiplication of vectors . . . . . . . . . . .
. . . . . . . . 1117 32 Differential and integral calculus 1121
32.1 Differential calculus . . . . . . . . . . . . . . . . . . . .
. . . . . . 1121 32.1.1 Differentiation rules . . . . . . . . . . .
. . . . . . . . . . . 1121 32.2 Integral calculus . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 1122 32.2.1 Integration rules
. . . . . . . . . . . . . . . . . . . . . . . . 1123 32.3
Derivatives and integrals of elementary functions . . . . . . . . .
. . . 1124 33 Tables on the SI 1125 Index 1131 Natural constants in
SI units 1183 Thermodynamic formulas 1184 Periodic table 1186
- 17. Contributors Mechanics: Christoph Best (Universitat
Frankfurt), with Helmut Kutz (Mauserwerke AG, Oberndorf) and
Rudolph Pitka, (Fachhochschule Frankfurt) Oscillations and Waves,
Acoustics, Optics: Kordt Griepenkerl (Universitat Frankfurt), with
Steffen Bohrmann (Fachhochschule Technik, Mannheim) and Klaus Horn
(Fachhochschule Frankfurt) Electricity, Magnetism: Christian
Hofmann, (Technische Universitat Dresden), with Klaus-Jurgen Lutz
(Universitat Frankfurt), Rudolph Taute (Fachhochschule der Telekom,
Berlin), and Georg Terlecki, (Fachhochschule Rheinland-Pfalz,
Kaiserslautern) Thermodynamics: Christoph Hartnack (Ecole de Mines
and Subatech, Nantes), with Jochen Gerber (Fachhochschule
Frankfurt), and Ludwig Neise (Universitat Heidelberg) Quantum
physics: Alexander Andreef (Technische Hochschule Dresden), with
Markus Hofmann (Universitat Frankfurt) and Christian Spieles
(Universitat Frankfurt) With contributions by: Hans Babovsky,
Technische Hochschule Ilmenau Heiner Heng, Physikalisches Institut,
Universitat Frankfurt Andre Jahns, Universitat Frankfurt Karl-Heinz
Kampert, Universitat Karlsruhe Ralf Rudiger Kories, Fachhochschule
der Telekom, Dieburg Imke Kruger-Wiedorn,
Naturwissenschaftliche-Technische Akademie Isny Christiane Lesny,
Universitat Frankfurt Monika Lutz, Fachhochschule Gieen-Friedberg
Raffaele Mattiello, Universitat Frankfurt Jorg Muller, University
of Tennessee, Knoxville Jurgen Muller, Denton Vacuum, Inc., and APD
Cryogenics, Inc. Frankfurt Gottfried Munzenberg, Universitat Gieen
and GSI Darmstadt xxiii
- 18. xxiv Contributors Helmut Oeschler, Technische Hochschule
Darmstadt Roland Reif, ehem. Technische Hochschule Dresden Joachim
Reinhardt, Universitat Frankfurt Hans-Georg Reusch, Universitat
Munster and IBM Wissenschaftliches Zentrum Heidelberg Matthias
Rosenstock, Universitat Frankfurt Wolfgang Schafer, Telenorma
(Bosch-Telekom) GmbH, Frankfurt Alwin Schempp, Institut fur
Angewandte Physik, Universitat Frankfurt Heinz Schmidt-Walter,
Fachhochschule der Telekom, Dieburg Bernd Schurmann, Siemens, AG,
Munchen Astrid Steidl, Naturwissenschaftliche-Technische Akademie,
Isny Jurgen Theis, Hoeschst, AG, Hochst Thomas Weis, Universitat
Dortmund Wolgang Wendt, Fachhochschule Technik, Esslingen Michael
Wiedorn, Gesamthochschule Essen und PSI Bern Bernd Wolf,
Physikalisches Institut, Universitat Frankfurt Dieter Zetsche,
Mercedes-Benz AG, Stuttgart We gratefully acknowledge numerous
contributions from textbooks by: Walter Greiner (Universitat
Frankfurt), and Werner Martienssen (Physikalisches Institut,
Universitat Frankfurt) The second edition included contributions
by: G. Brecht, FH Lippe, and DIN committee AEF H. Dirks, FH
Darmstadt E. Groth, FH Hamburg K. Grupen, Uni Siegen U. Gutsch, FH
Hanover S. Jordan, FH Schweinfurt P. Kienle, TU Munchen U. Kreibig,
Rheinisch-Westfalische Technische Hochschule, Aachen J.L.
Leichsenring, FH Koln H. Lockenhoff, FH Dortmund H. Merz, Uni
Munster J. Michele, FH Wilhemshaven H.D. Motz, Gesamthochschule
Wuppertal H. Niedrig, TH Berlin R. Nocker, FH Hanover H.J. Oberg,
FH Hamburg A. Richter, TH Darmstadt D. Riedel, FH Dusseldorf W.-D.
Ruf, FH Aalen J.A. Sahm, TU Berlin
- 19. Contributors xxv H. Schafer, FH Schmalkalden G. Zimmerer,
Uni Hamburg The third edition beneted from the efforts of: G. Flach
and N. Flach, who worked on format and illustrations R. Reif
(Dresden), who contributed to the sections on mechanics and nuclear
physics P. Ziesche (Dresden) and D. Lehmann (Dresden), who
contributed to the sections on condensed-matter physics J. Moisel
(Ulm), who contributed to the sections on optics R. Kories
(Dieburg), who contributed to the sections on semiconductor physics
E. Fischer (Arau), who provided detailed suggetions and a thorough
list of corrigenda for the second edition H.-R. Kissener, who
helped with the revisions of the entire book.
- 20. Part I Mechanics
- 21. 1 Kinematics Kinematics, the theory of the motion of
bodies. Kinematics deals with the mathematical description of
motion without considering the applied forces. The quantities
position, path, time, velocity and acceleration play central roles.
1.1 Description of motion Motion, the change of the position of a
body during a time interval. To describe the motion, numerical
values (coordinates) are assigned to the position of the body in a
coordinate system. The time variation of the coordinates
characterizes the motion. Uniform motion exists if the body moves
equal distances in equal time intervals. Op- posite: non-uniform
motion. 1.1.1 Reference systems 1. Dimension of spaces Dimension of
a space, the number of numerical values that are needed to
determine the position of a body in this space. A straight line is
one-dimensional, since one numerical value is needed to x the
position; an area is two-dimensional with two numerical values, and
ordinary space is three-dimensional, since three numerical values
are needed to x the position. Any point on Earth can be determined
by specifying its longitude and latitude. The dimension of Earths
surface is 2. The space in which we are moving is
three-dimensional. Motion in a plane is two- dimensional. Motion
along a rail is one-dimensional. Additional generalizations are a
point, which has zero dimensions, and the four-dimensional
space-time continuum (Minkowski space), the coordinates of which
are the three space coordinates and one time coordinate. 3
- 22. 4 1. Kinematics For constraints (e.g., guided motion along
rail or on a plane), the space dimension is restricted. 2.
Coordinate systems Coordinate systems are used for the mathematical
description of motion. They attach nu- merical values to the
positions of a body. A motion can thereby be described as a mathe-
matical function that gives the space coordinates of the body at
any time. There are various kinds of coordinate systems (ei : unit
vector along i-direction): a) Afne coordinate system, in the
two-dimensional case, two straight lines passing through a point O
(enclosed angle arbitrary) are the coordinate axes (Fig. 1.1); in
the three- dimensional case, the coordinate axes are three
different non-coplanar straight lines that pass through the
coordinate origin O. The coordinates , , of a point in space are
ob- tained as projections parallel to the three coordinate planes
that are spanned by any two coordinate axes onto the coordinate
axes. b) Cartesian coordinate system, special case of the afne
coordinate system, consists of respectively perpendicular straight
coordinate axes. The coordinates x, y, z of a space point P are the
orthogonal projections of the position of P onto these axes (Fig.
1.2). Line element: dr = dx ex + dy ey + dz ez. Areal element in
the x, yplane: dA = dx dy. Volume element: dV = dx dy dz. Figure
1.1: Afne coordinates in the plane, coordinates of the point P: 1,
1. Figure 1.2: Cartesian coordinates in three-dimensional space,
coordinates of the point P: x, y, z. Right-handed system, special
order of coordinate axes of a Cartesian coordinate system in
three-dimensional (3D) space: The x-, y- and z-axes in a
right-handed system point as thumb, forenger and middle nger of the
right hand (Fig. 1.3). c) Polar coordinate system in the plane,
Polar coordinates are the distance r from the origin and the angle
between the position vector and a reference direction (positive
x-axis) (Fig. 1.4). Line element: dr = dr er + r d e. Areal
element: dA = r dr d. d) Spherical coordinate system,
generalization of the polar coordinates to 3D space. Spherical
coordinates are the distance r from origin, the angle of the
position vector relative to the z-axis, and the angle between the
projection of the position vector onto the x-y-plane and the
positive x-axis (Fig. 1.5).
- 23. 1.1 Description of motion 5 Right-handed system Left-handed
system Figure 1.3: Right- and left-handed systems. Figure 1.4:
Polar coordinates in the plane. Coordinates of the point P: r, .
Line element: dr = dr er + r d e + r sin d e. Volume element: dV =
r2 sin dr d d. Spherical angle element: d = sin d d. e) Cylindrical
coordinate system, mixing of Cartesian and polar coordinates in 3D
space. Cylindrical coordinates are the projection (z) of the
position vector r onto the z- axis, and the polar coordinates (, )
in the plane perpendicular to the z-axis, i.e., the length of the
perpendicular to the z-axis, and the angle between this
perpendicular and the positive x-axis (Fig. 1.6). Line element: dr
= d e + d e + dz ez. Volume element: dV = d d dz. Figure 1.5:
Spherical coordinates. Figure 1.6: Cylindrical coordinates. 3.
Reference system A reference system consists of a system of
coordinates relative to which the position of the mechanical system
is given, and a clock indicating the time. The relation between the
reference system and physical processes is established by
assignment, i.e., by specication of reference points, reference
directions, or both. For a Cartesian coordinate system in two
dimensions (2D), one has to specify the origin and the orientation
of the x-axis. In three dimensions, the orientation of the y-axis
must also be specied. Alternatively, one can specify two or three
reference points.
- 24. 6 1. Kinematics There is no absolute reference system. Any
motion is a relative motion, i.e., it de- pends on the selected
reference system. The denition of an absolute motion without
specifying a reference system has no physical meaning. The
specication of the ref- erence system is absolutely necessary for
describing any motion. Any given motion can be described in many
different reference systems. The appro- priate choice of the
reference system is often a prerequisite for a simple treatment of
the motion. 4. Position vector and position function Position
vector, r, vector from the coordinate origin to the space point (x,
y, z). The posi- tion vector is written as a column vector with the
spatial coordinates as components: r = x y z . Position function,
r(t) = x(t) y(t) z(t) , species the position of a body at any time
t. The motion is denitely and completely described by the position
function. 5. Path Path, the set of all space points (positions)
that are traversed by the moving body. The path of a point mass
that is xed on a rotating wheel of radius R at the distance a <
R from the rotation axis, is a circle. If the wheel rolls on a at
surface, the point moves on a shortened cycloid (Fig. 1.7). Figure
1.7: Shortened cycloid as superposition of rotation and
translation. 6. Trajectory Trajectory, representation of the path
as function r(p) of a parameter p, which may be for instance the
elapsed time t or the path length s. With increasing parameter
value, the point mass runs along the path in the positive direction
(Fig. 1.8). Without knowledge of the time-dependent position
function, the velocity of the point mass cannot be determined from
the path alone. a) Example: Circular motion of a point mass. Motion
of a point mass on a circle of radius R in the x, y-plane of the 3D
space. Parametrization of the trajectory by the rotation angle as
function of time t: in spherical coordinates: r = R, = /2, = (t),
in Cartesian coordinates: x(t) = Rcos (t), y(t) = Rsin (t), z(t) =
0 (Fig. 1.9).
- 25. 1.1 Description of motion 7 Figure 1.8: Trajectory r(t).
Figure 1.9: Motion on a circle of radius R. Element of rotation
angle: , element of arc length: s = R . b) Example: Point on
rolling wheel. The trajectory of a point at the distance a < R
from the axis of a wheel (radius R) that rolls to the right with
constant velocity is a short- ened cycloid. The parameter
representation of a shortened cycloid in Cartesian coordinates in
terms of the rolling angle (t) (Fig. 1.10) reads: x(t) = vt a sin
(t), y(t) = R a cos (t). Figure 1.10: Parameter representation of
the motion on a shortened cycloid by the rolling angle as function
of time t. 7. Degrees of freedom of a mechanical system, number of
independent quantities that are needed to specify the position of a
system denitely. A point mass in 3D space has three translational
degrees of freedom (displacements in three independent directions
x, y, z). A free system of N mass points in 3D space has 3 N
degrees of freedom. If the motion within a system of N mass points
is restricted by inner or external constraints, so that there are k
auxiliary conditions between the coordinates r1, r2, . . . , rN ,
g(r1, r2, . . . , rN , t) = 0, = 1, 2, . . . , k , there remain
only f = 3 N k degrees of freedom with the system. For a point mass
that can move only in the x, y-plane (condition: z = 0), there
remain two degrees of freedom. The point mass has only one degree
of freedom if the motion is restricted to the x-axis (conditions: y
= 0, z = 0).
- 26. 8 1. Kinematics A system of two mass points that are
rigidly connected by a bar of length l has f = 6 1 = 5 degrees of
freedom (condition: (r1 r2)2 = l2, r1, r2: position vectors of the
mass points). A rigid body has six degrees of freedom: three
translational and three rotational. If a rigid body is xed in one
point (gyroscope), there remain three degrees of freedom of
rotation. A rigid body that can only rotate about a xed axis is a
physical pendulum with only one rotational degree of freedom. A
non-rigid continuous mass distribution (continuum model of a
deformable body) has innitely many degrees of freedom. 1.1.2 Time
1. Denition and measurement of time Time, t, for quantication of
processes varying with time. Periodic (recurring) processes in
nature are used to x the time unit. Time period, time interval, t,
the time distance of two events. M Time measurement by means of
clocks is based on periodic (pendulum, torsion vibra- tion) or
steady (formerly used: burning of a candle, water clock) processes
in nature. The pendulum has the advantage that its period T depends
only on its length l (and the local gravitational acceleration g):
T = 2 l/g. Mechanical watches use the periodic torsional motion of
the balance spring with the energy provided by a spiral spring.
Modern methods employ electric circuits in which the frequency is
stabilized by the resonance frequency of a quartz crystal, or by
atomic processes. Stopwatch, for measuring time intervals, often
connected to mechanical or electric devices for start and stop
(switch, light barrier). Typical precisions of clocks range from
minutes per day for mechanical clocks, over several tenths of
seconds per day for quartz clocks, to 1014 (one second in several
million years) for atomic clocks. 2. Time units Second, s, SI
(International System of Units) unit of time. One of the basic
units of the SI, dened as 9,192,631,770 periods of the
electromagnetic radiation from the transition between the hyperne
structure levels of the ground state of Cesium 133 (relative
accuracy: 1014). Originally dened as the fraction 864001 of a mean
solar day, subdivided into 24 hours, each hour comprising 60
minutes, and each minute comprising 60 seconds. The length of a day
is not sufciently constant to serve as a reference. [t] = s =
second Additional units: 1 minute (min) = 60 s 1 hour (h) = 60 min
= 3600 s 1 day (d) = 24 h = 1440 min = 86400 s 1 year (a) =
365.2425 d. The time standard is accessible by special radio
broadcasts. The Gregorian year has 365.2425 days and differs by
0.0003 days from the tropical year. Time is further divided into
weeks (7 days each) and months (28 to 31 days) (Gregorian
calendar).
- 27. 1.1 Description of motion 9 3. Calendar Calendar, serves
for further division of larger time periods. The calendar systems
are re- lated to the lunar cycle of ca. 28 days and to the solar
cycle of ca. 3651 4 days. Since these cycles are not commensurate
with each other, intercalary days must be included. Most of the
world uses the Gregorian calendar, which was substituted for the
former Julian calendar in 1582, at which time the intercalary rule
was modied for full century years. Since then, the rst day of
spring falls on March 20 or 21. The Julian calendar was in use in
eastern European countries until the October Rev- olution (1917) in
Russia. It differed from the Gregorian one by about three weeks.
Intercalary day, inserted at the end of February in all years
divisible by 4. Exception: full century years that are not
divisible by 400 (2000 is leap year, 1900 is not). Calendar week,
subdivision of the year into 52 or 53 weeks. The rst calendar week
of a year is the week that includes the rst Thursday of the year.
The rst weekday of the civil week is Monday, however it is Sunday
according to Christian tradition. Gregorian calendar years are
numbered consecutively by a date. Years before the year 1 are
denoted by B.C. (before Christ) or B.C.E. (before the Common Era to
Jews, Buddhists, and Muslims). There is no year Zero. The year 1
B.C. is directly followed by the year 1 A.D., or C.E. (Common Era)
Julian numbering of days: time scale in astronomy. Other calendar
systems: Other calendar systems presently used are the calendar
(luni- solar calendar, a mixture of solar and lunar calendar) that
involves years and leap months of different lengths; years are
counted beginning with 7 October 3761 B.C. (creation of the world)
and the year begins in September/October; the year 5759 began in
1998), and the Moslem calendar (purely lunar calendar with leap
month; years are counted beginning with the ight of Mohammed from
Mecca on July 16, 622 A.D.; the Moslem year 1419 began in the year
1998 of the Gregorian calendar). 1.1.3 Length, area, volume 1.
Length Length, l, the distance (shortest connecting line) between
two points in space. Meter, m, SI unit of length. One of the basic
units of the SI, dened as the distance traveled by light in vacuum
during 1/299792458 of a second (relative accuracy: 1014). The meter
was originally dened as the 40-millionth fraction of the
circumference of earth and is represented by a primary standard
made of platinum-iridium that is deposited in the Bureau
International des Poids et Mesures in Paris. [l] = m = meter.
Additional units see Tab. 33.0/3. 2. Length measurement Length
measurement was originally carried out by dening and copying the
unit of length (e.g., primary meter, tape measure, yardstick, screw
gauge, micrometer screw, often with a nonius scale for more
accurate reading). Interferometer: for precise optical measurement
of length (see p. 383) in which the wavelength of monochromatic
light is used as scale.
- 28. 10 1. Kinematics Sonar: for acoustical distance measurement
by time-of-ight measurement of ultra- sound for ships; used for
distance measurements with some cameras. Radar: for distance
measurement by means of time-of-ight measurement of electro-
magnetic waves reected by the object. Lengths can be measured with
a relative precision as good as 1014. Using micrometer screws, one
can reach precisions in the range of 106 m. Triangulation, a
geometric procedure for surveying. The remaining two edges of a
tri- angle can be evaluated if one edge and two angles are given.
Starting from a known basis length, arbitrary distances can be
measured by consecutive measurements of angles, using a theodolite.
Parallax, the difference of orientation for an object when it is
seen from two different points (Fig. 1.11). Applied to distance
measurement. Figure 1.11: Parallax for eyes separated by a distance
l and the object at a distance d: tan = l/d or l/d for d l. 3. Area
and volume Area A and volume V are quantities that are derived from
length measurement. Square meter, m2, SI unit of area. A square
meter is the area of a square with edge length of 1 m. [A] = m2 =
square meter. Cubic meter, m3, SI unit of volume. A cubic meter is
the volume of a cube with edge length 1 m. [V] = m3 = cubic meter.
M Areas can be measured by subdivision into simple geometric gures
(rectangles, tri- angles), the edges and angles of which are
measured (e.g., by triangulation), and then calculated. Direct area
measurement can be undertaken by counting the enclosed squares on a
measuring grid. Analogously, the volume of hollow spaces can be
evaluated by lling them with geometric bodies (cubes, pyramids, . .
. ). For the measurement of the volume of uids, one uses standard
vessels with known volume. The volume of solids can be determined
by submerging them in a uid (see p. 182). For a known density of a
homogeneous body, the volume V can be determined from the mass m, V
= m . Decimal prexes for area and volume units: The decimal prex
refers only to the length unit, not to the area or volume unit: 1
cubic centimeter = 1 cm3 = (1 cm)3 = 1 102 m 3 = 1 106 m3.
- 29. 1.1 Description of motion 11 1.1.4 Angle 1. Denition of
angle Angle, , a measure of the divergence between two straight
lines in a plane. An angle is formed by two straight lines (sides)
at their intersection point (vertex.) It is measured by marking on
both straight lines a distance (radius) from the vertex, and
determining the length of the arc of the circle connecting the
endpoints of the two distances (Fig. 1.12). angle and arc 1 = l r
Symbol Unit Quantity rad angle l m length of circular arc r m
radius Figure 1.12: Determination of the angle between the straight
lines g1 and g2 by measurement of the arc length l and radius r, l
= r . S: vertex 2. Angle units a) Radian, rad, SI unit of plane
angle. 1 rad is the angle for which the length of the circular arc
connecting the endpoints of the sides just coincides with the
length of a side. A full circle corresponds to the angle 2 rad.
Radian (and degree) are supplementary SI units, i.e., they have
unit dimensionality. 1 rad = 1 m/1 m. b) Degree, , also an accepted
unit for measurement of angles. A degree is dened as 1/360 of the
angle of a complete circle. Conversion: 1 rad = 360 2 = 57.3, 1 = 2
360 = 0.0175 rad. Subdivisions are: 1 degree () = 60 arc minutes (
) = 3600 arc seconds ( ). c) Gon (formerly new degree), a common
unit in surveying: 1 gon, 1/100 of a right angle. 1 gon = 0.9 =
0.0157 rad 1 = 1.11 gon 1 rad = 63.7 gon
- 30. 12 1. Kinematics M Measurement of angles: Measurement of
angles is performed directly by means of an angle scale, or by
measuring the chord of an angle and converting if the radius is
known. When determining distances by triangulation, the theodolite
(see p. 10) is used for angle measurement. 3. Solid angle Solid
angle, , is determined by the area of a unit sphere that is cut out
by a cone with the vertex in the center of the sphere (Fig. 1.13).
solid angle = A r2 Symbol Unit Quantity sr solid angle A m2 area
cut out by cone r m radius of sphere Figure 1.13: Determination of
the solid angle by measuring area A and radius r ( = A/r2).
Steradian, sr, SI unit of the solid angle. 1 steradian is the solid
angle that cuts out a surface area of 1 m2 on a sphere of radius 1
m (Fig. 1.14). This surface can be arbitrarily shaped and can also
consist of disconnected parts. The full spherical angle is 4 sr.
Radian and steradian are dimensionless. Figure 1.14: Denition of
the angular units radian (rad) (a) and steradian (sr) (b). The
(curved) area of the spherical segment A is given by A = 2 R h.
1.1.5 Mechanical systems 1. Point mass Point mass, idealization of
a body as a mathematical point with vanishing extension, but nite
mass. A point mass has no rotational degrees of freedom. When
treating the motion
- 31. 1.1 Description of motion 13 of a body, the model of point
mass can be used if it is sufcient under the given physical
conditions to study only the motion of the center of gravity of the
body, without taking the spatial distribution of its mass into
account. In the mathematical description of motion without
rotation, every rigid body can be replaced by a point mass located
in the center of gravity of the rigid body (see p. 94). For the
description of planetary motion in the solar system, it often
sufces to con- sider the planets as points, since their extensions
are very small compared with the typical distances between sun and
planets. 2. System of point masses System consisting of N
individual point masses 1, 2, . . . , N. Its motion can be
described by specifying the position vectors r1, r2, . . . , rN as
a function of the time t: ri (t), i = 1, 2, . . . N (Fig. 1.15a).
3. Forces in a system of point masses a) Internal forces, forces
acting between the particles of the system. Internal forces are in
general two-body forces (pair forces) that depend on the distances
(and possibly the velocities) of only two particles. b) External
forces, forces acting from the outside on the system. External
forces origi- nate from bodies that do not belong to the system. c)
Constraint reactions or reaction forces (external forces) result
from constraining the system. The interaction between the system
and the constraint is represented by reac- tions that act
perpendicularly to the enforced path. Constraint reactions restrict
the motion of the system. Guided motion: Mass on string xed at one
end, mass on an inclined plane, point mass on a straight rail,
bullet in a gun barrel. 4. Free and closed systems Free point mass,
free system of point masses, a point mass or a system of point
masses can react to the applied forces without constraints. Closed
system, a system that is not subject to external forces. 5. Rigid
body Rigid body, a body the material constituents of which are
always the same distances from each other, hence rigidly connected
to each other. For the distances of all points i, j of the rigid
body: |ri (t) rj (t)| = ri j = const. (Fig. 1.15b). Figure 1.15:
Mechanical systems. (a): system of N point masses, (b): rigid
body.
- 32. 14 1. Kinematics 6. Motion of rigid bodies Any motion of a
rigid body can be decomposed in two kinds of motion (Fig. 1.16): a)
Translation, all points of the body travel the same distance in the
same direction; the body is shifted in a parallel fashion. The
motion of the body can be described by the motion of a
representative point of the body. b) Rotation, when all points of
the body rotate about a common axis. Any point on the body keeps
its distance from the rotation axis and moves along a circular
path. Figure 1.16: Translation and rotation of a rigid body. (a):
translation, (b): rotation, (c): translation and rotation. 7.
Deformable body A deformable body can change its shape under the
inuence of forces. Described by many discrete point masses that are
connected by forces, or a continuum model according to which the
body occupies the space completely. 1.2 Motion in one dimension We
now consider motion along a straight-line path. The distance x of
the body from a xed point on the axis of motion is used as the
coordinate. The sign of x indicates on which side of the axis the
body is located. The choice of the positive x-axis is made by
convention. Position-time graph, graphical representation of the
motion (position function x(t)) of a point mass in two dimensions.
The horizontal axis shows the time t, the vertical axis the
position x (coordinate). 1.2.1 Velocity Velocity, a quantity that
characterizes the motion of a point mass at any time point. One
distinguishes between the mean velocity vx and the instantaneous
velocity vx . 1.2.1.1 Mean velocity 1. Denition of mean velocity
Mean velocity, vx , over a time interval t = 0, gives the ratio of
the path element x traveled during this time interval and the time
t needed (Fig. 1.17).
- 33. 1.2 Motion in one dimension 15 mean velocity = path element
time interval LT1 vx = x2 x1 t2 t1 = x(t1 + t) x(t1) (t1 + t) t1 =
x t Symbol Unit Quantity vx m/s mean velocity x1, x2 m position at
time t1, t2, resp. x(t) m position function t1, t2 s initial and
nal time point x m path element traveled t s time interval Figure
1.17: Mean velocity vx of one-dimensional motion in a position vs.
time graph. 2. Velocity unit Meter per second, ms1, the SI unit of
velocity. 1 m/s is the velocity of a body that travels one meter in
one second. A body that travels a distance of 100 m in one minute
has the mean velocity vx = x t = 100 m 60 s = 1.67 m/s. 3.
Measurement of velocity Velocity measurement can be performed by
time-of-ight measurement over a section of known length. Often it
is done by converting the translational motion into a rotational
one. Speedometer, for measuring speeds of cars. The rotational
motion of the wheels is trans- ferred by a shaft into the measuring
device where the pointer is moved by the centrifugal force arising
by this rotation (centrifugal force tachometer). In the
eddy-current speedometer, the rotational motion is transferred to a
magnet mounted in an aluminum drum on which the pointer is xed,
eddy currents create a torque that is balanced by a spring.
Electric speedometers are based on a pulse generator that yields
pulse sequences of higher or lower frequency corresponding to the
rotation velocity. Velocity measurement by Doppler effect (see p.
300) is possible using radar (automo- biles, airplanes, astronomy).
The velocity vx can have a positive or a negative sign,
corresponding to motion in either the positive or negative
coordinate direction. The mean velocity depends in general on the
time interval of measurement t. Ex- ception: motion with constant
velocity.
- 34. 16 1. Kinematics 1.2.1.2 Instantaneous velocity 1. Denition
of instantaneous velocity Instantaneous velocity, limit of the mean
velocity for time intervals approaching zero. instantaneous
velocity LT1 vx (t) = lim t0 x t = d dt x(t) = dx(t) dt = x(t)
Symbol Unit Quantity vx(t) m/s instantaneous velocity x(t) m
position at time t t s time interval x m path element The function
x(t) represents the position coordinate x of the point at any time
t. In the position-time graph, the instantaneous velocity vx(t) is
the slope of the tangent of x(t) at the point t (Fig. 1.18). The
following cases must be distinguished (the time interval t is
always positive): vx > 0: x > 0 and hence x(t + t) > x(t).
The body moves along the positive coordinate axis, i.e., the x-t
curve increases: the derivative of the curve x(t) is positive. vx =
0: x = 0 and hence x(t + t) = x(t), the distance x is constant
(zero). In this coordinate system the body is at rest (possibly
only briey), i.e., vx is the horizontal tangent to the x vs. t
curve, and the derivative of the curve x(t) vanishes. vx < 0: x
< 0 and hence x(t + t) < x(t). The body moves along the
negative coordinate axis, i.e., the x-t curve decreases, the
derivative of the curve x(t) is negative. 2. Velocity vs. time
graph Velocity vs. time graph, graphical representation of the
instantaneous velocity vx (t) as function of time t. To determine
the position function x(t) for a given velocity curve vx (t), the
motion is subdivided into small intervals t (Fig. 1.19). If the
interval from t1 to t2 is subdivided in N intervals of length t =
(t2 t1)/N, ti is the beginning of the ith time interval and vx(ti )
the mean velocity in this interval, then x(t2) = x(t1) + lim t0 N1
i=1 vx(ti ) t = x(t1) + t2 t1 vx (t) dt. path = denite integral of
the velocity over the time L x(t) = x(t1) + t t1 v() d x(t2) =
x(t1) + t2 t1 v(t) dt Symbol Unit Quantity x(t) m curve of motion
v(t) m/s velocity curve t1, t2 s beginning and ending time
points
- 35. 1.2 Motion in one dimension 17 Figure 1.18: Instantaneous
velocity vx at time t1 of one-dimensional motion in a position vs.
time graph. Figure 1.19: Velocity vs. time graph of one-dimensional
motion. ax: mean acceleration, ax : instantaneous acceleration at
time t1. 1.2.2 Acceleration Acceleration, the description of
non-uniform motion (motion in which the velocity varies). The
acceleration, as well as the velocity, can be positive or negative.
Both an increase (positive acceleration) and a decrease of velocity
(deceleration, as result of a deceleration process, negative
acceleration) are called acceleration. 1. Mean acceleration, ax ,
change of velocity during a time interval divided by the length of
the time interval: acceleration = change of velocity time interval
LT2 ax = vx t = vx2 vx1 t2 t1 Symbol Unit Quantity ax m/s2 mean
acceleration vx m/s velocity change t s time interval vx1, vx2 m/s
initial and nal velocity t1, t2 s initial and nal time Meter per
second squared, m/s2, SI unit of acceleration. 1 m/s2 is the
acceleration of a body that increases its velocity by 1 m/s per
second. If the mean acceleration and initial velocity are given,
the nal velocity reads vx2 = vx1 + ax t. The time needed to change
from the velocity vx1 to the velocity vx2 for given mean accel-
eration is t = vx2 vx1 ax . 2. Instantaneous acceleration
Instantaneous acceleration, limit of the mean acceleration for very
small time intervals ( t 0).
- 36. 18 1. Kinematics instantaneous acceleration LT2 ax (t) =
lim t0 vx t = dvx dt = d dt vx (t) Symbol Unit Quantity t s time
interval vx m/s velocity change ax (t) m/s2 acceleration vx (t) m/s
velocity The instantaneous acceleration ax (t) is the rst
derivative of the velocity function vx (t), and hence the second
derivative of the position function x(t): ax (t) = dvx(t) dt =
vx(t) = d dt dx(t) dt = d2x(t) dt2 = x(t). Graphically, it
represents the slope of the tangent in the velocity-time diagram
(Fig. 1.20). The following cases are to be distinguished: ax >
0: vx > 0 and hence vx2 > vx1. For vx1 > 0 the body moves
with increasing velocity, i.e., in the v vs. t graph the curve is
rising. ax = 0: vx = 0 and hence vx2 = vx1. The body does not
change its velocity (possibly only briey). ax < 0: vx < 0 and
hence vx2 < vx1. For vx1 > 0 the body moves with decreasing
velocity. Parabola Straight line Figure 1.20: Graphs for position
vs. time, velocity vs. time, and acceleration vs. time. Starting
from the origin, the body is rst uniformly accelerated, then moves
with constant velocity, and thereafter is uniformly decelerated to
rest. 3. Determination of velocity from acceleration If the
acceleration is given as function of time ax (t), the velocity is
determined by integra- tion: velocity = integral of acceleration
over time LT1 vx(t) = vx(t1) + t t1 a() d vx (t2) = vx(t1) + t2 t1
ax (t) dt Symbol Unit Quantity vx (t) m/s velocity curve ax (t)
m/s2 acceleration curve t1, t2 s initial and nal times
- 37. 1.2 Motion in one dimension 19 If a body has velocity v1x
< 0 and undergoes a positive acceleration ax > 0, the
velocity decreases in absolute value. 1.2.3 Simple motion in one
dimension Here we discuss uniform and uniformly accelerated motion
as the simplest forms of motion and discuss their physical
description. For motion in one dimension, one can omit the index x
and the vector arrow over the symbols for velocity v and
acceleration a. One should note, however, that v and a can take
positive and negative values and thus are components of vectors. 1.
Uniform motion Uniform motion, a motion in which the body does not
change its velocity, vx = vx = const. (Fig. 1.21). laws of uniform
motion x(t) = x0 + vx t vx(t) = vx = v0 ax(t) = 0 Symbol Unit
Quantity x(t) m position at time t x0 m initial position (t = 0) vx
m/s uniform velocity v0 m/s initial velocity t s time Figure 1.21:
Uniform motion. Uniform motion arises if no force acts on the body.
The curve of motion x(t) is the integral of the velocity curve vx
(t) = const. and is given by x(t) = x0 + t 0 vx(t ) dt = x0 + v0t.
Clearly, vx(t) is a straight line, and the integral corresponds to
the area below the straight between the points 0 and t on the time
axis. 2. Uniformly accelerated motion Uniformly accelerated motion,
a motion with constant acceleration. Then ax = ax = a and vx (t) =
at + v0, if v0 is the initial velocity (Fig. 1.22).
- 38. 20 1. Kinematics Figure 1.22: Uniformly accelerated motion.
It follows by integration that x(t) = t 0 vx (t ) dt + x0 = t 0 (at
+ v0) dt + x0 = a 2 t2 + v0t + x0. This result can also be read
from the velocity vs. time graph: the area below the curve is
composed of a rectangle of area v0 t and a triangle of area at2/2
(height at and basis t) (Fig. 1.23). Figure 1.23: Graphs for
uniformly accelerated motion. uniformly accelerated motion x(t) = a
2 t2 + v0t + x0 vx(t) = at + v0 ax(t) = a = const. Symbol Unit
Quantity x(t) m position at time t vx(t) m/s velocity t s